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A Note on Bessel’s Inequality

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Dragomir, Sever S (2001) A Note on Bessel’s Inequality. RGMIA research report collection, 4 (2).

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S.S. DRAGOMIR

Abstract. A monotonicity property of Bessel’s inequality in inner product spaces is given.

1. Introduction

Let X be a linear space over the real or complex number fieldK. A mapping (·,·) :X×X →Kis said to be apositive hermitian formif the following conditions are satisfied:

(i) (αx+βy, z) =α(x, z) +β(y, z) for allx, y, z∈X andα, β∈K; (ii) (y, x) = (x, y) for all x, y∈X;

(iii) (x, x)≥0 for allx∈X.

If kxk := (x, x)¹² , x∈ X denotes the semi-norm associated to this form and (e_i)_i∈I is an orthornormal family of vectors inX, i.e., (e_i, e_j) =δ_ij (i, j∈I), then one has the following inequality [15]:

(1.1) kxk²≥X

i∈I

|(x, ei)|² for allx∈X, which is well known in the literature as Bessel’s inequality.

Indeed, for every finite partH ofI, one has:

0 ≤

x−X

i∈H

(x, e_i)e_i

2

=



x−X

i∈H

(x, e_i)e_i, x−X

j∈H

(x, e_j)e_j





= kxk²−X

i∈H

|(x, ei)|²−X

j∈H

|(x, ej)|²+ X

i,j∈H

(x, ei) (ej, x)δij

= kxk²−X

i∈H

|(x, ei)|², for allx∈X,which proves the assertion.

The main aim of this paper is to improve this result as follows.

2. Results The following theorem holds.

Theorem 1. Let X be a linear space and (·,·)₂,(·,·)₁ two hermitian forms on X such that k·k₂ is greater than or equal to k·k₁, i.e., kxk₂ ≥ kxk₁ for all x ∈ X. Assume that(ei)_i∈I is an orthornormal family in(X; (·,·)₂)and(fi)_i∈J is an

Date: January 05, 2000.

1991Mathematics Subject Classification. Primary 26D15; Secondary 46C99.

Key words and phrases. Bessel’s inequality, Inner product spaces.

1

2 S.S. DRAGOMIR

orthornormal family in(X; (·,·)₁)such that for anyi∈Ithere exists a finiteK⊂J so that

(F) ei =X

j∈K

αjfj, αj ∈K (j∈K), then one has the inequality:

(2.1) kxk²₂−X

i∈I

|(x, e_i)₂|²≥ kxk²₁−X

j∈J

(x, f_j)₁

2≥0,

for allx∈X.

In order to prove this fact, we require the following lemma.

Lemma 1. Let X be a linear space endowed with a positive hermitian form (·,·) and(g_k)_k=1,n be an orthornormal family in(X; (·,·)). Then

(2.2)

x−

n

X

k=1

λkgk

2

≥ kxk²−

n

X

k=1

|(x, gk)|²≥0, for allλk∈Kandx∈X (k= 1, ..., n).

Proof. We will prove this fact by induction over “n”.

Supposen= 1. Then we must prove that

kx−λ1g1k²≥ kxk²− |(x, g1)|², x∈X, λ1∈K.

A simple computation shows that the above inequality is equivalent with

|λ1|²−2 Re (x, λ1g1) +|(x, g1)|²≥0, x∈X, λ1∈K. Since Re (x, λ1g1)≤ |(x, λ1g1)|, one has

|λ₁|²−2 Re (x, λ₁g₁) +|(x, g₁)|² ≥ |λ₁|²−2|λ₁| |(x, g₁)|+|(x, g₁)|²

≥ (|λ1| − |(x, g1)|)²≥0 for allλ1∈Kandx∈X, which proves the statement.

Now, assume that (2.2) is valid for “(n−1)”. Then we have:

x−

n

X

k=1

λkgk

2

=

(x−λngn)−

n−1

X

k=1

λkgk

≥ kx−λngnk²−

n−1

X

k=1

|(x−λngn, gk)|²

= kx−λ_ng_nk²−

n−1

X

k=1

|(x, g_k)|²≥ kxk²− |(x, g_n)|²−

n−1

X

k=1

|(x, g_k)|²

= kxk²−

n

X

k=1

|(x, gk)|²,

for allλk ∈K,x∈X (k= 1, ..., n),and the proof of the lemma is complete.

Proof. (Theorem) LetH be a finite part of I. Sincek·k₂ is greater than k·k₁, we have:

kxk²₂−X

i∈H

|(x, e_i)₂|² =

x−X

i∈H

(x, e_i)₂e_i

2

2

≥

x−X

i∈H

(x, e_i)₂e_i

2

1

, x∈X.

Since, by (F), we may state that for anyi∈H there exists a finiteK⊂J with ei= X

j∈K

(ei, fj)₁fj, we have

x−X

i∈H

(x, e_i)₂e_i

2

1

=

x−X

i∈H

(x, e_i)₂X

j∈K

(e_i, f_j)₁f_j

2

1

=

x−X

j∈K

X

i∈H

(x, ei)₂ei, fj

!

1

fj

2

1

for allx∈X.

Applying the above lemma for (·,·) = (·,·)₁, (g_k)_k=1,n = (f_j)_j∈K, we can conclude that

x−X

j∈K

λjfj

2

1

≥ kxk²₁−X

j∈K

(x, fj)₁

2, x∈X,

where

λj = X

i∈H

(x, ei)₂ei, fj

!

1

∈K (j∈K). Consequently, we have:

kxk²₂−X

i∈H

|(x, ei)₂|²≥ kxk²₁−X

j∈K

(x, f_j)₁

2≥ kxk²₁−X

j∈J

(x, f_j)₁

2

for allx∈X andH a finite part ofI, from where results (2.1).

The proof is thus completed.

Corollary 1. Let k·k₁,k·k₂:X →R+be as above. Then for allx, y∈X, we have the inequality:

(2.3) kxk²₂kyk²₂− |(x, y)₂|²≥ kxk²₁kyk²₁− |(x, y)₁|²≥0, which is an improvement of the well known Cauchy-Scwartz inequality.

Proof. Ifkyk₂= 0,then (2.3) holds with equality.

If kyk_i 6= 0, (i= 1,2), then for {e1} = n _y

kyk₂

o

, {f1} = n _y

kyk₁

o

, the above theorem yields that

kxk²₂kyk²₂− |(x, y)₂|²

kyk²₂ ≥ kxk²₁kyk²₁− |(x, y)₁|² kyk²₁

4 S.S. DRAGOMIR

and sincekyk₂≥ kyk₁,the inequality (2.3) is obtained.

Remark 1. For a different proof of (2.3), see also [5].

Now, we will give some natural applications of the above theorem.

3. Applications

(1) Let (X; (·,·)) be an inner product space and (ei)_i∈I an orthornormal family in X. Assume that A:X →X is a linear operator such thatkAxk ≤ kxk for all x ∈ X and (Ae_i, Ae_j) = δ_ij for all i, j ∈ I. Then one has the inequality

kxk²−X

i∈I

|(x, ei)|²≥ kAxk²−X

i∈I

|(Ax, Aei)|²≥0 for allx∈X.

The proof follows by the hermitian forms (x, y)₂ = (x, y) and (x, y)₁ = (Ax, Ay) forx, y∈X and for the family (fi)_i∈I = (ei)_i∈I.

(2) If A : X → X is such that kAxk ≥ kxk for all x ∈ X, then, with the previous assumptions, we also have

0≤ kxk²−X

i∈I

|(x, ei)|²≤ kAxk²−X

i∈I

|(Ax, Aei)|², for allx∈X.

(3) Suppose that A : X → X is a symmetric positive definite operator with (Ax, x) ≥ kxk² for all x∈ X. If (ei)_i∈I is an orthornormal family in X such that (Aei, Aej) =δij for alli, j∈I, then one has the inequality

0≤ kxk²−X

i∈I

|(x, ei)|²≤(Ax, x)−X

i∈I

|(Ax, ei)|², for allx∈X.

The proof follows from the above theorem for the choices (x, y)₁= (Ax, y) and (x, y)₂= (x, y), x, y∈X. We omit the details.

For other inequalities in inner product spaces, see the papers [1]-[14] and [7]-[6]

where further references are given.

References

[1] S.S. DRAGOMIR, A refinement of Cauchy-Schwartz inequality, G.M. Metod.(Bucharest), 8(1987), 94-95.

[2] S.S. DRAGOMIR, Some refinements of Cauchy-Schwartz inequality,ibid,10(1989), 93-95.

[3] S.S. DRAGOMIR and B. MOND, On the Boas-Bellman generalisation of Bessel’s inequality in inner product spaces,Italian J. of Pure and Appl. Math.,3(1998), 29-38.

[4] S.S. DRAGOMIR and B. MOND, On the superadditivity and monotonicity of Gram’s in- equality and related results,Acta Math. Hungarica,71(1-2) (1996), 75-90.

[5] S.S. DRAGOMIR and B. MOND, On the superadditivity and monotonicity of Schwartz’s inequality in inner product spaces, Contributions Macedonian Acad. Sci. and Arts,15(2) (1994), 5-22.

[6] S.S. DRAGOMIR, B. MOND and Z. PALES, On a supermultiplicity property of Gram’s determinant,Aequationes Mathematicae,54(1997), 199-204.

[7] S.S. DRAGOMIR, B. MOND and J.E. PE ˇCARI ´C, Some remarks on Bessel’s inequality in inner product spaces,Studia Univ. “Babes-Bolyai”, Math.,37(4) (1992), 77-86.

[8] S.S. DRAGOMIR and J. SANDOR, On Bessel’s and Gram’s inequalities in prehilbertian spaces,Periodica Math. Hungarica,29(3) (1994), 197-205.

[9] S.S. DRAGOMIR and J. S ´ANDOR, Some inequalities in prehilbertian spaces,Studia Univ.

“Babe¸s-Bolyai”,Mathematica, 1,32, (1987), 71-78.

[10] W.N. EVERITT, Inequalities for Gram determinants, Quart. J. Math., Oxford, Ser. (2), 8(1957), 191-196.

[11] T. FURUTA, An elementary proof of Hadamard theorem,Math. Vesnik,8(23)(1971), 267- 269.

[12] C.F. METCALF, A Bessel-Schwartz inequality for Gramians and related bounds for deter- minants,Ann. Math. Pura Appl.,(4)68(1965), 201-232.

[13] D.S. MITRINOVI ´C,Analytic Inequalities, Springer-Verlag, Berlin-Heidelberg and New York, 1970.

[14] C.F. MOPPERT, On the Gram determinant,Quart. J. Math., Oxford, Ser (2), 10(1959), 161-164.

[15] K. YOSHIDA,Functional Analysis, Springer-Verlag, Berlin, 1966.

School of Communications and Informatics, Victoria University of Technology, P.O.

Box 14428, Melbourne City MC, Victioria 8001, Australia E-mail address:sever.dragomir@vu.edu.au

URL:http://rgmia.vu.edu.au/SSDragomirWeb.html